In this problem, we are asked to identify which of the three distributions are normal and which of the two normal distributions has a larger variation (or standard deviation).

Step 1: Identifying Normal Distributions

A normal distribution is characterized by a smooth, bell-shaped curve that is symmetric around its peak. Let's analyze the descriptions of the three curves:

• Curve (a): A smooth curve begins just above the origin, rises gradually to a maximum, and falls gradually to just above the x-axis.

  • Observation: This sounds like a bell-shaped, symmetric curve. Therefore, (a) is a normal distribution.

• Curve (b): A smooth curve begins at the origin, rises sharply to a maximum, and falls gradually to just above the x-axis.

  • Observation: This is also a bell-shaped curve, though with one side rising more steeply than the other. Despite the asymmetry in how it rises and falls, the description suggests a bell-shaped curve, so (b) is also a normal distribution.

• Curve (c): A smooth curve begins at the origin, rises sharply to a maximum, and falls sharply to the x-axis.

  • Observation: This is not a normal distribution. The rise and fall are both steep, which resembles a non-symmetric distribution such as a triangular or other skewed distribution. Therefore, (c) is not a normal distribution.

Thus, the two normal distributions are (a) and (b).

Step 2: Comparing Variation

The standard deviation (or variation) of a distribution indicates the spread of the data. A curve with a larger spread will have a larger standard deviation. We can compare the descriptions of curves (a) and (b):

• Curve (a): Rises gradually and falls gradually.
• Curve (b): Rises sharply but falls gradually.

Since curve (a) rises and falls more gradually, it has a wider spread, implying a larger standard deviation compared to curve (b), which rises more sharply.

Conclusion:

• The two normal distributions are (a) and (b).
• Curve (a) has the larger standard deviation.

Would you like more details or have any questions? Here are 5 related questions to explore further:

1. How is the standard deviation of a normal distribution calculated?
2. What are the key properties of a normal distribution?
3. How can we distinguish between normal and non-normal distributions in real data?
4. How does the shape of a curve affect the interpretation of its standard deviation?
5. What are examples of real-world phenomena that follow normal distributions?

Tip: In a normal distribution, about 68% of the data falls within one standard deviation of the mean, and 95% falls within two standard deviations.